The Hebrew text explicitly says, "because the Lord has anointed me." So it is not the Holy Spirit who anoints, but He is the anointing oil itself, in Jesus' case. I would argue that Jesus was anointed with and by the Holy Spirit Himself, sent by the Father (John 14:26), to preach the gospel by His power while on earth, and not by His own divine power as separate (cf. Luke 11:20).
The identification of the Messiah or Anointed King, the oil of anointing, and the Spirit, are concepts tied together in other Scriptures, such as:
1 Kings 16:6-13 And when they were come in, he saw Eliab, and said: Is the Lord's anointed before him? 7 And the Lord said to Samuel: Look not on his countenance, nor on the height of his stature: because I have rejected him, nor do I judge according to the look of man: for man seeth those things that appear, but the Lord beholdeth the heart. 8 And Isai called Abinadab, and brought him before Samuel. And he said: Neither hath the Lord chosen this. 9 And Isai brought Samma, and he said of him: Neither hath the Lord chosen this. 10 Isai therefore brought his seven sons before Samuel: and Samuel said to Isai: The Lord hath not chosen any one of these. 11 And Samuel said to Isai: Are here all thy sons? He answered: There remaineth yet a young one, who keepeth the sheep. And Samuel said to Isai: Send, and fetch him, for we will not sit down till he come hither. 12 He sent therefore and brought him. Now he was ruddy and beautiful to behold, and of a comely face. And the Lord said: Arise, and anoint him, for this is he. 13 Then Samuel took the horn of oil, and anointed him in the midst of his brethren: and the spirit of the Lord came upon David from that day forward: and Samuel rose up, and went to Ramatha.
It seems that in Jesus' case, He was sent out with His Messianic ministry, and officially declared King (the Anointed), by the Holy Spirit directly, without need for oil (which merely symbolized the Spirit's coming).
Matthew 3:13-17 Then cometh Jesus from Galilee to the Jordan, unto John, to be baptized by him. 14 But John stayed him, saying: I ought to be baptized by thee, and comest thou to me? 15 And Jesus answering, said to him: Suffer it to be so now. For so it becometh us to fulfill all justice. Then he suffered him. 16 And Jesus being baptized, forthwith came out of the water: and lo, the heavens were opened to him: and he saw the Spirit of God descending as a dove, and coming upon him. 17 And behold a voice from heaven, saying: This is my beloved Son, in whom I am well pleased.
This shows, in my opinion, that Jesus is above the need for sacrament or sign, and is favored by God by natural right. This is different from His circumcision or dedication (or redemption) at the Temple, since He was utterly dependant on the knowledge of His parents then, and the transition from Old to New Covenants was not yet complete.